Mortality profit

[ivanapavone]

How would one actually calculate the mortality profit for the year if we were considering, let’s say, a monthly annuity?

Would it still be correct to use the recursive formula for the reserves and isolate the “qx” part in order to get the DSAR ( this is where the UDD assumption comes in and allows me to express relevant terms in terms of qx)?

Also because I’m assuming UDD, that would imply that people die on average halfway through the year, so would the ADS be the value of the reserve at exactly half way through the year if I have no date of death?

 

[Em Francis]

Hi
You want to get the DSAR to the end of the year. So if the only benefit was monthly annuities then, using the assumption that deaths are halfway through the year, the survival benefit 'R' would be 6 months worth of annuity payments which you could then assume are paid halfway through the half year, ie three quarters of the way through the year and so you end up ...

with a DSAR = -tV - (-0.5*annual annuity amount*(1+i)^0.25)

The EDS is then : number of policies* qx * DSAR
And the ADS = number of deaths * DSAR

Thanks
Em

 

[ivanapavone]

Would it be more correct to account for the fact that the monthly annuity payments that are paid in the first half of the year are disinvested and so don’t earn interest from the time they are disinvested up until the end of the year for the DSAR?

 

[Em Francis]

Would it be more correct to account for the fact that the monthly annuity payments that are paid in the first half of the year are disinvested and so don’t earn interest from the time they are disinvested up until the end of the year for the DSAR?

But we are assuming that deaths are half way through the year, and so the first half of the year's annuities are paid?

 

[ivanapavone]

But we are assuming that deaths are half way through the year, and so the first half of the year's annuities are paid?

But because it’s paid monthly, should we take the different timing of cash flows into account? Essentially, as they are paid, they have the opportunity cost of not earning the investment returns over the year.

 

[Joe Hook]

Fundamentally we want to compare actual death strain to expected death strain (additional money actually required over and above our reserves vs what we expected). In the case of a monthly annuity we'd ideally work monthly with actual deaths vs expected deaths (with some assumption about the mortality rate in that month). We could then sum up the positions across the year to get the total mortality profit. My expectation is that in real-life problems we would be able to calculate this and so anything else is purely hypothetical.

If we're going to work yearly then we're in approximation territory however you look at it, and i'd expect any reasonable assumption would do. The key as Em suggests is to put everything on a consistent (end-year) basis. So we have:

t+1V - the reserve we'd hold if everyone survived
S - the sum assured on death (0)
R - the survival benefits "foregone" if a life dies during the year, which would be on average half a years payments (the second half a years) accumulated to the end of the year.

Because the payments in the first half of the year would be made to those that survive and those that die (on average) there's no contribution to profit/loss there in my mind?